How the Math Gets Done

Catheryne Draper has been learning from her students for over half a century of teaching, supervising the math program in a school district, advising math education at the state level, coaching math in schools, and presenting math workshops for teachers. She is the author of The Algebra Game, a hands-on multi-deck algebra program in four topics covering Linear Graphs, Quadratic Equations, Conic Sections, and Trig Functions that allows students to work together in cooperative groups, or individually, to identify the algebra relationships and patterns in the each topic and in the organization across the topics. In addition to contributing many published articles, Draper is also the author of Winning the Math Homework Challenge: Insights for Parents To See Math Differently and User-Friendly Math for Parents: Learning and Understanding the Language of Numbers is Key.

Foreword
Preface
Introduction
Part I. Definition
1. Math According to Jen, Bobby, and Others
Jen’s Deductions
Bobby’s Experience with Multiplication and Number Arrangements
Children’s Descriptions About Making Sense
Trevor and Jim Invent Their Own Multiplication Methods
Keep in Mind
2. Math Grammar of Nouns, Verbs, and Stories
“Putting Together” Verbs for Addition and Multiplication
“Taking Apart” Verbs for Subtraction and Division
A Short Interlude About the Use of that Negative Sign
Word Problems Put Math Verbs in Stories
Keep in Mind
3. A Choice between Two Rs – Rote Memorization or Reasoning
Sound Bites That Bite Back
Memorization vs. Organization
Acronyms - Sense or Nonsense
Keep in Mind
Part II. Organization
4. In Search of Like Terms, Classification Revisited
Like Terms, Common Denominators, and Same Units
Place Value Columns Have Like Terms Categories
What’s Wrong with this Picture?
Adding Percentages
A Note About Multiplication and Division
Keep in Mind
5. Artful Assembly of Operations
Facts, Frogs, and Formats
Jackie’s Thinking About Number Operations
James, Janie, and Napier’s Lattice Multiplication
Division Interpretations That Made Sense
Keep in Mind
6. Same Math, Same Meaning, Different Organization – New vs. Old
“If It Ain’t Broke, Don’t Fix It.”
Multiplication Organization Sense
Digit Alignment Continues
Zeke, Jake, and Long Division
Keep in Mind
Part III. Relationships and Patterns
7. “Seeing” Math Patterns with Viewfinders
Multiplication Table Patterns
Addition Table Patterns
Kathy’s and Rudy’s Fraction Viewfinders
Keep in Mind
8. Spelunking for Patterns with More Viewfinders
Reflections on Addition and Multiplication
Subtraction, Division, and Missy’s “Different Family”
A Few Rules of Engagement for Working with Numbers
Keep in Mind
9. Functions, Predictability, and Balance
Predictability and Functions
One Answer or Many Answers, Same Balance
Keep in Mind
Part IV. Connections
10. A Multiplication Area Image for the Ages
From Theon to Dienes – A Bridge Across the Centuries
Same Design for Mixed Numbers, Fractions and Decimals
Visual-Spatial Multi-Digit Multiplication
Keep in Mind
11. Ratios, Proportions, and Rate of Change
Fractions as Gatekeepers
David’s Decluttering Fractions
Changing Numbers, Changing Locations, and Moving Targets in Proportions
Slope as a Rate of Change
Keep in Mind
12. Algebraic Thinking
The Shapes for an Algebra Transition
From Multiplication Tables to Coordinate Tables
Jill’s Graphic Solutions
F.O.I.L.’ed Again
Keep in Mind
Conclusion: What Parents Can Do
Problem Solving Beyond Word Problems
Right Tool for the Right Job - A Hard Look At Technology
Keep in Mind
Glossary
References
About the Author